# Evolutionarily stable strategy

Conflicts between organisms are often modelled as games (cf. also Games, theory of). Here, attention is confined to conflicts between con-specifics, which are also symmetric in the sense that each player has available the same set of strategies (cf. Strategy (in game theory)) and the same pay-off function. (Important texts in this area are [a5], [a8], [a15], [a17], [a22].)

Consider first two-player conflicts (cf. also Two-person zero-sum game). Suppose there exists a set of possible strategies , and a pay-off function , where , the pay-off to a player who plays against one who plays . One extends the set of available strategies to some , where each is a weight function defined on the elements of ; e.g. a probability function if is discrete. One supposes that the extension of the pay-off function is linear, i.e. for ,

where denotes summation or integration (as appropriate) and is the weight attached to by .

A concept of solution of an evolutionarily stable strategy was introduced in [a19]. Such a strategy, if adopted by the whole of the (infinite) population (see [a21] for the finite case), cannot be invaded by any alternate strategy introduced at low frequency.

A strategy is said to be evolutionarily stable with respect to if:

1) ;

2) implies .

A strategy is said to be evolutionarily stable if is evolutionarily stable with respect to for all . For a discussion of the definition see also [a5] and [a23]. Let denote the support of . If is such that for all while for all , then is a strict Nash equilibrium (cf. also Nash theorem (in game theory)). It follows that , so is evolutionarily stable against for all such that , being the empty set. Thus, in the generic case an evolutionarily stable strategy is a strict Nash equilibrium with the additional condition that for all such that one requires that , or, equivalently, that for all such that one requires that , with equality if and only if .

For conditions for to be an evolutionarily stable strategy in the finite (i.e. matrix) case, see [a10] (generic) and [a1] (non-generic). If is an evolutionarily stable strategy and is such that , then is not an evolutionarily stable strategy (the Bishop–Cannings theorem, [a4]).

The war of attrition was introduced in [a16], where and and if , if and if . Thus, individuals contest for a reward by choosing a single value, perhaps display time, the winner being the one choosing the larger value but the cost to each only being the lesser; i.e. the contest ends immediately when the first individual retreats. In this case there is a unique evolutionarily stable strategy, given by , , where . This follows since , , with equality if and only if , which ensures that is an evolutionarily stable strategy, and this is unique by the Bishop–Cannings theorem. If , i.e. individuals essentially commit themselves irrevocable to their choice, as might occur for a choice of size, then there is no evolutionarily stable strategy [a18]. A hybrid model with has been treated by [a12].

## Contents

## Finite conflicts.

As stated above, [a10] gives conditions in the finite (i.e. matrix) case for to be an evolutionarily stable strategy. Suppose and . For , an evolutionarily stable strategy with exists if and only if and . See, in particular, [a19] for the hawk–dove scenario, where there is a reward for winning, a hawk always beats a dove, doves share the reward, while hawks share the reward but also incur damage (cost ). Thus,

and there is an evolutionarily stable strategy with both hawk and dove if , demonstrating that organisms need not adopt aggressive strategies. On the other hand, if , then is an evolutionarily stable strategy, while if , then is an evolutionarily stable strategy. Thus there is always at least one evolutionarily stable strategy.

For , [a10] has given examples of the possible sets of evolutionarily stable strategies, it being possible that there be no evolutionarily stable strategy. More generally, suppose that , where , denoting the power set. Then is called a pattern. If there exists an -matrix with evolutionarily stable strategies with supports , then is said to be attainable on . The Bishop–Cannings theorem ensures that an attainable pattern is a Sperner family. Various restrictions on the set of attainable patterns have been derived in [a24]; in it, it is conjectured that if is attainable and , then is also attainable on . This conjecture is still (1996) open, although in [a6] it has been proved that is attainable for some .

## Dynamics.

A major criticisism of the evolutionarily stable strategy solution concept is that it is a static, rather than a dynamic concept. However, for the continuous replicator dynamics, if there is an interior evolutionarily stable strategy , then convergence to that evolutionarily stable strategy is assured, where [a26], and, more generally, if is such that is a convex combination of some subset of its elements, then convergence is assured to the linear manifold defined by [a2]. If the players can choose itself, then convergence is assured even for more general dynamics [a7]. For further discussion of dynamics see [a14], which uses a covariance approach and for an examination of the embedding of such dynamics within a genetic system, and a discussion of whether evolution can find the evolutionarily stable strategy, see [a9], [a13], [a25].

## Iterated conflicts, spatial conflicts and cooperation.

The classic paradigm for examining the evolution of cooperation is the prisoner dilemma [a3] (see [a22] for an excellent overview). Here the defect strategy dominates the cooperate strategy (). However, if the conflict consists not of a single play, but of a series of random (geometrically distributed) length, then cooperation can evolve. A series of strategies (tit-for-tat, i.e. imitate the opponent's last play, two-tits-for-tat, generous-tit-for-tat, etc.) has been developed, and although none of these is an evolutionarily stable strategy, studies suggest that cooperation, in some form, may evolve. In a related way, spatial structure (players are located at the points of a grid, interact with some set of neighbours and imitate the best local strategy in the next round) may lead to the persistence of both strategies [a20].

## Multi-player conflicts.

A multi-player war of attrition model is discussed in [a11]. This necessitates the extension of the notion of an evolutionarily stable strategy to players. Assume that the pay-off for playing against opponents depends only on the membership of the set of strategies and not on any ordering within the set. Thus, the pay-off to an individual who plays while of the opponents play and play can be written as . A strategy is said to be evolutionarily stable with respect to in a -player conflict if there exists a such that

1) , for ;

2) . An evolutionarily stable strategy is then defined as for the two-player conflict in terms of evolutionary stability. Note that the definition, while only referring to two strategies, implicitly deals with the case of several strategies. There are then situations in which no evolutionarily stable strategy exists in the war of attrition, e.g. if players are allowed to re-assess their play when some other player quits. It is clear that the study of multi-player conflicts and their application to the establishment and maintenance of dominance hierarchies, to mate selection, etc., is an area of considerable importance.

#### References

[a1] | A. Abakucs, "Conditions for evolutionarily stable strategies" J. Appl. Probab. , 17 (1980) pp. 559–562 |

[a2] | E. Akin, "Exponential families and game dynamics" Canadian J. Math. , 34 (1982) pp. 374–405 |

[a3] | R. Axelrod, "Further evolution of co-operation" Science , 242 (1988) pp. 1385–1390 |

[a4] | D.T. Bishop, C. Cannings, "A generalised war of attrition" J. Theor. Biol , 70 (1978) pp. 85–124 DOI 10.1016/0022-5193(78)90304-1 |

[a5] | I.M. Bomze, B.M. Potschner, "Game theoretical foundations of evolutionary stability" , Lecture Notes Econom. and Math. Systems , Springer (1998) |

[a6] | M. Broom, C. Cannings, G.T. Vickers, "Sequential methods in patterns of ESS's" J. Math. Biol. , 22 (1994) pp. 597–615 |

[a7] | C. Cannings, "Topics in the theory of ESS's" S. Lessard (ed.) , Proc. NATO Advanced Study Institute (Montreal, 1987) , Kluwer Acad. Publ. (1990) |

[a8] | R. Cressman, "The stability concept of evolutionary game theory" , Springer (1992) |

[a9] | I. Eshel, M.W. Feldman, "Initial increase of new mutants and continuity properties of ESS in two-locus systems" Amer. Nat. , 124 (1984) pp. 631–640 |

[a10] | J. Haigh, "Game theory and evolution" Advances Appl. Probab. , 7 (1975) pp. 8–10 |

[a11] | J. Haigh, C. Cannings, "The $n$-person war of attrition" Acta Applic. Math. , 14 (1989) pp. 59–74 |

[a12] | J. Haigh, M.R. Rose, "Evolutionary game auctions" J. Theor. Biol. , 85 (1980) pp. 381–397 |

[a13] | P. Hammerstein, "Darwinian adaptation, population genetics and the streetcar theory of evolution" J. Math. Biol. , 34 (1996) pp. 511–532 |

[a14] | W.G.S. Hines, D. Anfossi, "A discusson of evolutionarily stable strategies" S. Lessard (ed.) , Proc. NATO Adv. Study Inst. (Montreal, 1987) , Kluwer Acad. Publ. (1990) |

[a15] | J. Hofbauer, K. Sigmund, "The theory of evolution and dynamical systems" , Cambridge Univ. Press (1988) |

[a16] | J. Maynard Smith, "The theory of games and the evolution of animal conflicts" J. Theor. Biol. , ,47 (1974) pp. 209–221 DOI 10.1016/0022-5193(74)90110-6 |

[a17] | J. Maynard Smith, "Evolution and the theory of games" , Cambridge Univ. Press (1982) |

[a18] | J. Maynard Smith, G.E. Parker, "The logic of asymmetric conflicts" Animal Behavior , 24 (1976) pp. 159–175 |

[a19] | J. Maynard Smith, G.R. Price, "The logic of animal conflict" Nature , 246 (1973) pp. 15–18 |

[a20] | M.A. Nowak, R.M. May, "Evolutionary games and spatial chaos" Nature , 359 (1992) pp. 826–829 |

[a21] | J.G. Riley, "Evolutionary equilibrium strategies" J. Theor. Biol. , 76 (1979) pp. 109–123 DOI 10.1016/0022-5193(79)90365-5 |

[a22] | K. Sigmund, "Games of life" , Oxford Univ. Press (1993) |

[a23] | G.T. Vickers, C. Cannings, "On the definition of an evolutionarily stable strategy" J. Theor. Biol. , 129 (1987) pp. 349–353 |

[a24] | G.T. Vickers, C. Cannings, "Patterns of ESS's 1" J. Theor. Biol. , 132 (1988) pp. 387–408 |

[a25] | F.J. Weissing, "Genetic versus phenotypic models of selection: Can genetics be neglected in a long term perspective?" J. Math. Biol. , 34 (1996) pp. 533–555 |

[a26] | E.C. Zeeman, "Population dynamics from game theory" A. Nitecki (ed.) C. Robinson (ed.) , Global Theory of Dynamical Systems , Springer (1980) |

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Evolutionarily stable strategy.

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